Chaotic movement



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Introduction


Throughout its history, the human species has been interested in knowing any type of mobility existing in its environment, starting from the movements that we have called bases such as circular, parabolic, elliptical and hyperbolic, and these represent fundamental mobilities for the rest of this type of phenomena that are generally of a combined or even more complex type.
In this opportunity we will continue extracting from our environment more movements as we have done with the phenomena such as the oscillatory, vibratory and undulatory, expanding the set or family of this type of manifestations either naturally or artificially, highlighting the path followed which has represented the geometric place of different shapes or geometric figures.
In this article we will analyze the so-called chaotic movement, and when we refer to that word (chaos) we immediately think of difficulties, however, the scientific field has already accustomed us to the fact that some words implemented in that area do not have the same meaning in ordinary language, and therefore, these words usually have abstract characteristics for everyday language.
Therefore, we can express that the word chaos represents one of these examples, and science has made it its own to evidence or demonstrate an extreme sensitivity related to the initial conditions of certain dynamic systems, both natural and mechanical non-linear.
It is important to emphasize that when we refer to the sensitivity in the initial conditions of a certain dynamic system, we are expressing that any two points of this system could have mobility through very different trajectories one with respect to the other, even if this difference with respect to the initial conditions is very small in their respective space of phases.
The phase space represents the whole set of possible configurations for a specific system, for example, to analyze the kinematic aspect of a pendulum we could determine a configuration for it by knowing both the angular velocity (ω) and the angle (α) of the movement, this is because one would not be enough, and for this case the phase space would have two dimensions.
The study of the sensitivity with respect to the initial conditions of a certain dynamic system is represented by the historical example of the butterfly effect, where the delicate flapping of the wings of this small lepidopteran could originate sensitive or delicate changes to our atmosphere and these could be transformed over time into any phenomenon as violent and devastating as a storm, then we observe the sensitive flapping of a butterfly in the following figure 1.

It is important to bear in mind that the main objective of chaos theory is to show that certain dynamic systems, both natural and artificial, when subjected to some type of changes, however small in their initial conditions, would lead the system to experience great divergences or differences in the results of its dynamics, such as the one described above with the butterfly effect.
Next we will show a natural phenomenon of great force that is originated by these small changes in its initial conditions as it is observed in the following figure 2.

Our environment as well as our lives are linked to such phenomenon because we live immersed in chaos, the first researcher of this phenomenon was the meteorologist Edward Lorentz when he used a mathematical model with the purpose of predicting the weather, where, he used a computer for the simulation, and in this he obtained an approximation or probable behavior of the atmosphere.
However, when he wanted to repeat these computations introducing the values obtained in the computer and using 3 decimals instead of 6 as he initially did, for this case he could notice that this result was totally different from the one calculated previously, therefore, this description led us to the theory that we know today as the chaos theory.
Highlighting again the fundamental idea of this principle, which expresses, that to very small differences in relation to the initial conditions of a certain dynamic system, these differences had great impacts in the final result of a certain dynamic analysis of any system, then, as we expressed previously to this characteristic of the small initial differences we know them as butterfly effect.
Lorentz's concern in discovering the above was based on the fact that it was impossible to accurately predict or foresee the respective behavior of any system, and this was due to the failure or error of the instruments at the time of making the calculations and consequently all the measurements would be affected, and therefore it would be impossible to accurately determine the initial conditions of most dynamic systems.
In spite of the above, Lorentz could notice that the solution of those systems which seemed to behave as a totally random fact, after observing them as graphs, that is, through curves or geometric representations he could conclude that such result always represented a certain region of space and which acquired the geometric shape or figure of a double spiral.
When detecting this behavior of the dynamic system observed I visualize the solution of the system, these solutions are always ordered since they draw a spiral and they will not stop in a point, neither they will repeat and neither they will be periodic, this geometric representation is called Lorentz's Atractor, and this one must also fulfill the condition of not cutting itself since if this happens it would originate two different curves in this cut point, next we will observe the trajectory before described in the Lorentz's Atractor in the following figure 3.

It is vital to be able to highlight again some important characteristics of this type of chaotic systems, since they will always be very sensitive to the initial conditions, so a very small change in their respective initial data will give rise to widely divergent results, these systems seem to be disordered or products of chance, but they definitely are not, because they are governed by certain rules that model their behavior, highlighting then that any system produced at random would not represent a chaotic system.
Having clear that a chaotic movement is very complex and impossible to predict, the description of the chaos in our body or organism represents one of the clearest examples of such chaotic systems, and this is because it would be impossible to predict the movement that any particle will have internally in our system of organs which make up the human body, among many other events related to the particular evolution of any organism of a particular person.

Chaotic Movement

In our daily life we can say that we implement the term chaos when we refer to a type of disorder of any kind, however, in physical science this expression is implemented to give a sense of behavior to those systems both natural and artificial which evolve in a certain time and depending extremely on their initial conditions.
During the development of both our history and our intellectual capacity we have been able to notice that any physical system has the capacity to evolve in some way over time, and whose evolution will be linked to the initial conditions of that system. In general, in a dynamic system its initial conditions will be determined both by the position and by the speed of each particle that makes it up.
A given dynamic system, for example, consisting of a particle attached to a spring and which is moved away from its position of stable equilibrium at a distance (A), being at that initial point at rest or motionless, will evolve by performing a Simple Harmonic Motion whose amplitude will clearly be determined by the value of the magnitude (A).
In a simple mass-spring system, by modifying even slightly the initial position of the particle attached to the spring, this will cause a slight change in the amplitude of the referred movement, however, it will continue to be a Simple Harmonic Movement, therefore, we could say that the modification of the initial position of the particle will be proportional to the resulting amplitude for such movement, that is, in this case a small variation will cause a small evolution of the dynamic system mentioned above.
But we do not always find this type of simple systems, since there are more complex systems where small variations in their initial conditions will cause great variations in the evolution of that system, so this type of more complex systems which are strongly dependent on their initial conditions we know as chaotic systems, then we will see an example of a chaotic system in the following figure 4.

The double pendulum represents a clear example of this type of chaotic mobility and it is constituted or formed by two (2) coupled masses and its chaotic effect can be better observed when the referred ligatures (l1 and l2) are rigid and when a considerable force is applied or supplied to them, their oscillations will be longer in terms of their amplitude, which leads to unpredictable movements.
This chaotic behavior increases when implementing a pendulum of three (3) masses and therefore three (3) ligatures, preferably rigid, the more changes in their initial conditions the greater their chaotic mobilities, this type of pendulum can be observed in the following figure 5.

Conclusion

We can express that any type of system, either natural or artificial has the possibility of evolving in relation to time, that is, they have one or more properties that change over time, the various variables that would be analyzed in any system could have any value other than zero, this at the time that the magnitude of time (t) begins to time, therefore these variables are what we know as initial conditions.
According to the above we arrived at the point where we wanted to arrive, the chaotic movement, any of us would easily or spontaneously relate the meaning of chaos with clear disorder this from an everyday point of view, however, we could express that one of the most important characteristics of the concept of chaos would be the lack of predictability, that is, to be able to know when it will happen.
For any dynamic system to tend to be chaotic it has to be related to two very important essential characteristics; one of them is represented by the non-linearity and the other by the high sensitivity related to the changes of the initial conditions, therefore, we can express that when altering the conditions in relation to the initial time, the result of a certain chaotic system is unpredictable.
So far we have shown that when we talk about complexity we are referring to the phenomenon of movement, and in fact we know that it will always be that way since that characteristic is transmitted by our natural environment, since everything around us depends essentially on some kind of movement, either in a particular way going through a specific trajectory such as a straight line, circumference, parabola, ellipse and hyperbola, but also going through a combination of these.
The phenomenon of movement accompanies us everywhere, and the chaotic movement is not the exception, as we could observe in each one of the previous figures or examples, therefore, any natural or artificial system can develop this type of phenomenon when being altered its condition by very small that is this alteration, taking us to the effect butterfly and the theory of the chaos.
Until another opportunity my appreciated friends and readers of steemit, very specially to the members of the big communities friends of #steemstem and #curie, for which I recommend widely to be part of these exemplary projects, because they highlight the valuable work of the academy and the scientific field.
Note: All images were made using the applications, Power Point and the animated gif with the PhotoScape.

Bibliographic References

[1]Charles H. Lehmann. Analytic geometry [2]Serway Raymond, Emeritus James Madison. Physics for Scientists and Engineers. 6th edition [3]Crystal Ives. Human Beings as Chaotic Systems [4]Garnett P. Williams. Chaos theory Tamed [5]Hateley James. The Lorenz system

Comments 9


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16.01.2020 16:21
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Hello,

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16.01.2020 21:40
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Thank you for your valuable support. Greetings.

19.01.2020 19:30
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17.01.2020 03:14
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Upvoted 👌 (Mana: 5/10 - need recharge?)

17.01.2020 03:14
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Thank you brother @ritch for your valuable support. Greetings.

19.01.2020 19:32
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Those mechanisms have more than one degree of freedom thus we cannot find the instant center of rotation for some of the bars, in that case, we cannot describe their movement because we do not know where is such point. The movement of 1 degree of freedom mechanisms can be described with precision, for example, the movement of the mass m1. Great article, greetings! @acont

17.01.2020 14:19
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Thanks for your valuable contribution @acont it is great to find people like you with great analytical skills. A brotherly greeting, my friend.

19.01.2020 19:28
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